Speed of convergence of Chernoff approximations to solutions of evolution equations

Abstract

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations.Introduction. Since the middle of the XX century it is a well known fact [1,2] that the solution of a well-posed Cauchy problem for a linear evolution partial differential equation (examples: Schödinger equation, parabolic equations) is given by a strongly continuous semigroup of linear bounded operators whose infinitesimal generator is a (usually unbounded) linear operator from the right-hand side of the evolution equation. Let us explain this in more details and introduce some notation which will be useful for the main text.Let X be an infinite set, and F be a Banach space of (not necessarily all) number-valued functions on X, and let L be a closed linear operator L : Dom(L) → F with the domain Dom(L) ⊂ F dense in F . We consider the Cauchy problem for the evolution equation